窄带及有界噪声激励下非线性粘弹性系统的可靠性及其最优控制研究

Viscoelastic materials are widely used in mechanical engineering, civil engineering, aerospace engineering, electric engineering, bioengineering and other areas as its particular energy storage and damping functions. The response and stability of the viscoelastic system subjected to random excitations have been extensively investigated. Nevertheless, the reliability of the stochastic system, which is the equally important topic and the vital index of the life and safety evaluation of the original system, has been rarely researched except for the situation of the white or the wideband noise excited system. For the viscoelastic system under the narrowband or the bounded noise, which is common in the engineering practice, the study of the reliability is almost a blank. In addition, the prediction of reliability for stochastic viscoelastic system under narrowband or bounded noise has important theoretical guiding significance and engineering application values. In the present project, reliability of SDOF or MDOF nonlinear viscoelastic systems excited by narrowband or bounded noises is studied. The main procedures are as follows. Firstly, the viscoelastic term is approximately replaced by equivalent damping and stiffness separately. By using the stochastic averaging method based on the generalized harmonic functions, the averaged Itô stochastic differential equation with respect to system amplitude is obtained. The associated backward Kolmogorov equation governing the conditional reliability function is derived and the solving of this backward equation yields the numerical solution of system reliability. Secondly, the modified Laplace integral method is applied to acquire the approximate analytical solution of the reliability and the mean first passage time of the viscoelastic system. Then, the dynamical programming principle on the averaged system is utilized to obtain the optimal control strategy for maximizing system reliability. Finally, the accuracy of the evaluation results and the applicability of approximate analytical method are substantiated by the Monte Carlo simulations.

粘弹性材料因其特殊的储能和减振性能，广泛应用于机械、土木、航空、电子及生物工程等领域。随机激励下的粘弹性系统的响应与稳定性已被广泛研究，然而对于其可靠性问题的研究仅局限于白噪声或宽带噪声激励情形，对工程中常见的窄带及有界噪声情形的研究几乎为空白。可靠性是评估系统寿命及安全性能的重要指标，因此研究窄带及有界噪声激励下粘弹性系统的可靠性具有重要的理论意义及工程应用价值。本项目以单/多自由度非线性粘弹性系统为研究对象，研究窄带及有界噪声激励下粘弹性系统的可靠性。基本研究方法是：将粘弹性项近似为等效刚度和阻尼项，利用基于广义谐和函数的随机平均法，得到关于系统幅值的伊藤方程及后向柯尔莫戈洛夫方程，并用于求解系统可靠性的数值解；运用修正的拉普拉斯积分方法，得到系统可靠性的近似解析解；利用动态规划原理，得到提高系统可靠性的最优控制；引入蒙特卡洛模拟，验证研究结果的准确性及所用近似解析方法的适用范围。

粘弹性材料由于其特殊属性而被广泛应用。非线性粘弹性系统的随机响应和随机稳定性已经得到深入研究，然而对于同样重要的可靠性问题，目前的研究还局限于平稳随机激励下的粘弹性系统的数值解。本项目基于广义Maxwell模型的经典本构方程，利用随机平均法和修正的Laplace积分法研究了非平稳随机激励下非线性粘弹性系统的可靠性及其平均首次穿越时间。结合随机平均法与差分法，求解了谐和与白噪声联合激励下的单自由度粘弹性系统，得到了该系统的随机响应与条件可靠性函数。将单自由度系统的研究方法推广到多自由度，推导了窄带及有界噪声激励下多自由度粘弹性系统的平均Itô方程、后向Kolmogorov方程及Pontryagin方程，并得到了系统的条件可靠性函数与平均首次穿越时间的数值解。对于非高斯、非宽带噪声激励下的粘弹性系统，将Laplace积分法进行修正，再结合粘弹性系统的等效近似、随机平均法、差分法或超松弛迭代法，得到了谐和与白噪声联合激励下单自由度粘弹性系统首次穿越问题的近似解析解。将该方法应用于多自由度系统，发现得到的结果与Monte Carlo模拟结果及差分法得到的结果误差较大，判断出使用的修正Laplace积分法对于多自由度非线性系统适用条件较为苛刻。此外，对于随机系统的响应或者可靠性问题，Monte Carlo模拟主要用于验证所提出方法的准确性。然而，在工程实际中，Monte Carlo法通常指的是一种随机抽样方法，为了探索该方法的实际意义，项目团队将其应用于汽车座椅鞭打试验的不确定度评价当中。本项目分析影响座椅鞭打评分项的输入输出参数，并研究其概率密度分布情况；利用基于神经网络和支持向量机的机器学习方法，建立输入输出参数之间的数学模型，并与试验结果和有限元仿真结果对比，验证模型的精度；用Monte Carlo法对输入参数随机抽样，并导入到建立好的数学模型中，预测出输出参数；根据不确定度评价规则和鞭打试验评分准则，得到座椅鞭打试验的不确定度。最后，研究过程中，考虑到理论与实际相结合，项目团队正搭建振动系统的实验装置，用于验证理论和仿真研究结果。